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Vinayak Joshi

Publications and source records attributed to Vinayak Joshi.

17 recordsLinked to original sources
arXiv2026

On $S$-prime and $S$-primary elements in multiplicative lattices

In this paper, we study $S$-prime elements and $S$-primary elements within the framework of multiplicative lattices. Furthermore, we define and explore weakly $S$-prime elements and weakly $S$-primary elements, which generalize weakly prime elements and weakly primary elements in multiplicative lattices respectively. We show that the weakly $S$-prime ideals (weakly $S$-primary ideals) of a commutative ring $R$ with $1$ correspond precisely to the weakly $S_L$-prime elements (weakly $S$-primary elements) of the ideal lattice $Id(R)$ of $R$, where $S_L = \{(s) \mid s \in S\}$.

math.AC↗
arXiv2026

On Isomorphism theorem of the Comparability Graph of Lattices

In recent years, researchers have actively contributed to the field of graphs associated with algebraic structures and ordered structures. It is a fundamental question to ask whether we infer algebraic or ordered structure from associated graphs and vice versa. In this paper, we gave characterizations about comparability graphs and associated lattices. In particular, we determined some properties of lattices that are preserved under the graph isomorphism. We have also provided a technique to construct non-isomorphic lattices with isomorphic comparability graphs. Also, we find two classes of lattices in which the graph isomorphism gives the lattice isomorphism.

math.CO↗
arXiv2026

On $S$-Noetherian Lattices

In this paper, we define and study $S$-Noetherian lattices as a natural generalization of Noetherian rings. We prove that a ring $R$ is $S$-Noetherian if and only if its ideal lattice, $Id(R)$, is $S_L$-Noetherian. Furthermore, we establish a Cohen-Kaplansky type theorem for $S$-Noetherian lattices, showing that $L$ is $S$-Noetherian if and only if every $S$-prime element of $L$ is $S$-compact. Finally, we introduce the concept of $S$-primary elements-a generalization of primary elements in multiplicative lattices and demonstrate the existence and uniqueness of $S$-primary decomposition in $S$-Noetherian lattices.

math.AC↗
arXiv2026

On $S$-Prime Element Principle

In this paper, we introduce $S$-prime elements in $V$-lattices, where $S$ is a multiplicatively closed subset of a $V$-lattice $L$. In addition, we introduce the $S$-Prime Element Principle to prove that certain elements in $V$-lattices are $S$-prime elements. This principle leads to a direct and uniform approach to the results on the existence of prime elements in multiplicative lattices when $S=\{1\}$.

math.AC↗
arXiv2026

Complemented zero-divisor graph of posets

In this paper, we derive a set of equivalent conditions for the zero-divisor graph $Γ(Q)$ of a poset $Q$ with $0$ to be complemented, characterizing it in terms of quasi-complemented posets. Furthermore, we prove that the notions of a complemented zero-divisor graph and a uniquely complemented zero-divisor graph coincide for any poset $Q$ with $0$. In addition, we provide both algebraic and topological characterizations for $Γ(Q)$ to be a complemented graph. In the final section, we apply these characterizations to the zero-divisor graphs of a reduced (multiplicative) semigroup $S$ with $0$ and the comaximal (ideal) graph of an Artinian ring $R$, and the nonzero component union graph $\mathbb{UG}(\mathbb{V})$ of a finite-dimensional vector space $\mathbb{V}$ over a field $\mathbb{F}$.

math.CO↗
arXiv2026

A Proof of the Conjecture on complemented zero-divisor graphs of semigroups

In this paper, we are motivated by the conjectures proposed by C.~Bender \textit{et al.}, \cite{C} in 2024. We have settled the first two conjectures negatively by providing a counter example in \cite{KTJ}, whereas in this paper, we prove the third conjecture positively, which has remained an open question until now. The third conjecture is stated as if $G(S)$ is uniquely complemented with the clique number $3$ or greater and has the property that every vertex has a unique complement, then the graph $G(S)$ is isomorphic to the graph $G(\mathcal{P}(n))$, where $n$ is the clique number of $G(S)$.

math.CO↗
arXiv2025

Thermodynamic Phase Transitions and Quantum Entropy Corrections in the Simpson-Visser Regular Black Hole

Regular black holes offer a compelling framework to explore the consequences of resolving the central singularity of standard black holes. Using the Simpson-Visser ``black-bounce" geometry as an elegant, analytically tractable framework, we explore the intricate thermodynamic behavior in such models. We demonstrate that this regular spacetime exhibits a critical instability, marked by a phase transition where the heat capacity is discontinuous. This transition signals a fundamental change in the black hole's evaporation state, which depends on the regularization parameter. Pushing beyond the semiclassical limit, we then derive the leading-order quantum corrections to the entropy via the Hamilton-Jacobi tunneling formalism. Our analysis provides a refined statistical basis for the entropy of non-singular spacetimes and offers a quantitative analysis of the nature of the black hole end-state. These results reveal that singularity resolution is not merely a geometric modification but a profound thermodynamic event, with direct implications for the stability and ultimate fate of evaporating black holes.

gr-qc↗
arXiv2025

On the strong metric dimension of the complement of the zero-divisor graph of a lattice

In this paper, we compute the strong metric dimension of the complement of the zero-divisor graph of the blow-up of a Boolean lattice. Using these results, we calculate the strong metric dimension of the total graph, the maximal graph, the intersection graph of ideals, the complement of the zero-divisor graph of a reduced ring, and the component graph of a vector space.

math.CO↗
arXiv2025

Counter-example to Conjectures on Complemented Zero-Divisor Graphs of Semigroups

In this paper, we are motivated by two conjectures proposed by C. Bender et al.\ in 2024, which have remained open questions. The first conjecture states that if the complemented zero-divisor graph \( G(S) \) of a commutative semigroup \( S \) with a zero element has the clique number three or greater, then the reduced graph \( G_r(S) \) is isomorphic to the graph \( G(\mathcal{P}(n)) \). The second conjecture asserts that if \( G(S) \) is a complemented zero-divisor graph with the clique number three or greater, then \( G(S) \) is uniquely complemented. In this work, we construct a commutative semigroup \( S \) with a zero element that serves as a counter-example to both conjectures.

math.CO↗
arXiv2024

On the strong metric dimension of the zero-divisor graph of a lattice

In this paper, the generalized blow-up of a Boolean lattice $L\cong \textbf{2}^n$ using finite chains is introduced. Also, we compute the strong metric dimension of the zero-divisor graph of the blow-up of a Boolean lattice. These results are applied to calculate the strong metric dimension of the comaximal graph, the comaximal ideal graph, the zero-divisor graph of a reduced ring, and the component graph of a vector space.

math.CO↗
arXiv2022

Component graphs of vector spaces and zero-divisor graphs of ordered sets

In this paper, nonzero component graphs and nonzero component union graphs of finite dimensional vector space are studied using the zero-divisor graph of specially constructed 0-1-distributive lattice and the zero-divisor graph of rings. Further, we define an equivalence relation on nonzero component graphs and nonzero component union graphs to deduce that these graphs are the graph join of zero-divisor graphs of Boolean algebras and complete graphs. In the last section, we characterize the perfect and chordal nonzero component graphs and nonzero component union graphs.

math.CO↗
arXiv2022

Coloring of zero-divisor graphs of posets and applications to graphs associated with algebraic structures

In this paper, we characterize chordal and perfect zero-divisor graphs of finite posets. Also, it is proved that the zero-divisor graphs of finite posets and the complement of zero-divisor graphs of finite $0$-distributive posets satisfy the Total Coloring Conjecture. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graph of rings, the annihilating ideal graphs, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of cyclic groups. In fact, it is proved that these graphs associated with a commutative ring $R$ with identity can be effectively studied via the zero-divisor graph of a specially constructed poset from $R$.

math.CO↗
arXiv2021

$\mathfrak{X}$-elements in multiplicative lattices -- A generalization of $J$-ideals, $n$-ideals and $r$-ideals in rings

In this paper, we introduce a concept of $\mathfrak{X}$-element with respect to an $M$-closed set $\mathfrak{X}$ in multiplicative lattices and study properties of $\mathfrak{X}$-elements. For a particular $M$-closed subset $\mathfrak{X}$, we define the concept of $r$-element, $n$-element and $J$-element. These elements generalize the notion of $r$-ideals, $n$-ideals and $J$-ideals of a commutative ring with unity to multiplicative lattices. In fact, we prove that an ideal $I$ of a commutative ring $R$ with unity is a $n$-ideal ($J$-ideal) of $R$ if and only if it is an $n$-element ($J$-element) of $Id(R)$, the ideal lattice of $R$.

math.AC↗
arXiv2016

Zero-divisor graphs of lower dismantlable lattices-II

In this paper, we continue our study of the zero-divisor graphs of lower dismantlable lattices that was started in [20]. The present paper mainly deals with an Isomorphism Problem for the zero-divisor graphs of lattices. In fact, we prove that the zero-divisor graphs of lower dismantlable lattices with the greatest element 1 as join-reducible are isomorphic if and only if the lattices are isomorphic.

math.CO↗
arXiv2013

Beck's Conjecture For Multiplicative Lattices

In this paper, we introduce the zero divisor graph of a multiplicative lattice. We provide a counter example to Beck's conjecture for multiplicative lattices. Further, we prove that Beck's conjecture is true for reduced multiplicative lattice which extends the result of Behboodi and Rakeei[7], and Aalipour et. al.[1].

math.AC↗
arXiv2013

Diameter and Girth of Zero Divisor Graph of Multiplicative Lattices

In this paper, we study the zero divisor graph $Γ^m(L)$ of a multiplicative lattice L. We prove under certain conditions that for a reduced multiplicative lattice L having more than two minimal prime elements, $Γ^m(L)$ contains a cycle and $gr(Γ^m(L)) = 3$. This essentially proves that for a reduced ring R with more than two minimal primes, $gr(\mathbb{AG}(R))) = 3$ which settles the conjecture of Behboodi and Rakeei [9]. Further, we have characterized the diameter of $Γ^m(L)$.

math.AC↗